**Lester R Ford**was involved in mathematics training for soldiers. His course was published in 1919 as:

**Elementary Mathematics for Field Artillery.**

By Lester R Ford.

(Prepared and published by direction of the Chief of Field Artillery.) Field Artillery Central Officers' Training School, Camp Zachary Taylor, Kentucky (Instruction Memo. No. 20). Louisville, Kentucky (Courier Journal Press, 1919). 72 pages.

The following review by Albert A Bennett, University of Texas, is interesting both in its own right and also for what it tells us of Ford's course:

Educators have recently become critical. Before deciding how a subject ought to be taught, it is now customary to question whether it ought to be taught at all, and to admit that the aim to be achieved may have an influence upon the methods to be employed. Elementary mathematics has been under fire. in many quarters, and while the word "mathematics" has long ceased to be used as a plural noun efforts are but now being exerted in numerous directions to impress upon the pupil the unity of the science, particularly in coordinated freshman courses.

The war presented an exceptional although fleeting opportunity to emphasize the fundamental necessity of mathematics in the technical equipment of the individual. This was most conspicuous perhaps in the case of artillery and naval units. The student officer was required to learn and use many mathematical ideas that he had never acquired, or had frequently long forgotten. The result at least in many of the training camps was rather pitiable. The instructors themselves usually knew the formulas, but not the reasoning, and blind confusion was the rule rather than the exception among the students. In the S.A.T.C. program, the situation was nearly as bad. Even had classes not been interrupted, the students would never have acquired many of the essentials, for two reasons: much of the time was spent with unnecessary details, and again the instructors were not prepared in many cases to give what was needed. They did not know what concepts were employed, what problems were to be faced, and frequently what were the mathematical facts of the subjects even were these named. Such terms as mil, grad, deflection, azimuth, dispersion diagram, probable error, centre of impact, nomogram, or alignment chart ' meant little or nothing to many college instructors in mathematics, while most college graduates could not approximate a square root by the algorithm, determine visibility on a map with the contour lines given, interpolate with second differences, and the like. But all of these things had to be taught, not to a few only but to many. Not having been given even in a college course, these topics had to be learned from men of action rather than teachers, from soldiers who had never been concerned with mathematical theory.

It was in the face of an emergency of this sort that the present pamphlet was prepared and issued, and although the military urgency has passed there is much suggestiveness in this little text-book, for the teacher of collegiate mathematics. The reviewer has learned that "about 15,000 students took the course in the three months prior to the signing of the armistice. The staff of instructors, recruited chiefly from the candidates who had mathematical training, was ordinarily well over a hundred, and at one time numbered as many as one hundred sixty-nine."

This pamphlet comprises sections entitled "arithmetic," "algebra," "geometry," "trigonometry," "approximate methods," "coordinates," "aids to calculation," and "probability," and has also an appendix of tables.

In the section on arithmetic the most interesting features emphasized are the conversion of metric into English units and vice versa, various units of angular measure, mil, grad, degree, the process of extracting the square root, and interpolation. The algebra and geometry sections cover customary topics; the geometry section is particularly rich in theorems, but many of the proofs are omitted. Under trigonometry, seven pages altogether, the sine, cosine, tangent, and cotangent are mentioned, but not the secant or cosecant. The law of sines and the law of cosines are derived and applied, but triangles with three sides given are not discussed and the law of tangents is not mentioned. Under "aids to calculation," logarithms, the slide rule, and a nomographic logarithmic table are explained. Under the topic of probability, the probability curve and the dispersion diagram are illustrated and applied. The tables are to four decimal places, and comprise (I) square roots of numbers, (II) logarithms of numbers, (III) natural trigonometric tables-mils, (IV) logarithmic trigonometric tables - mils, (V) natural and logarithmic tables - degrees.

This memorandum as an official publication contains of course neither preface nor introduction, but was obviously written with a single purpose in view-to serve as a text in teaching the field artillery officer the mathematics he must know. No topic that is necessary is omitted on account of its difficulty, nor is any topic of clearly minor importance for this object introduced because of its symmetry or historical interest. Nearly every problem mentioned in the exercises bears on its face the answer to the common query, "But why must I study this?" The work is written for adults, and is so condensed as to serve well for a reference book and source of problems but scarcely as a guide for self instruction of the mentally deficient. From a logical point of view, the book is surprisingly satisfactory. The definitions are clear and cover the terms used, the arrangement is progressive, where proofs are omitted there is no attempt to conceal the fact, processes are always explained in general terms, as well as applied to numerical examples, emphasis is placed on the operations used rather than on formulas-in short the book is written by a mathematician. In his experience in teaching from American text books designed for standard first year-college courses, the reviewer has seldom found a treatment with so few logically objectionable elements; of the numerous mathematical memoranda he has seen, used in the army training schools, this is the first which shows ease and clarity of treatment.

A few of the exercises may be cited:

- Near the muzzle of the three-inch gun the rifling makes one turn per 25 calibers. If the projectile leaves the muzzle with a velocity of 1,700 feet per second, how many rotations about its axis does it make per second?

Ans. 272. - The total length of the French 155 mm. cannon is 2.332 m. Express this in calibers.

Ans. 15.05. - In rising above the earth's surface the temperature falls about 6° C for each kilometer of altitude up to eleven kilometers and thereafter remains practically stationary. What increase in elevation (to the nearest ten feet) will lower the temperature 1° F?

Ans. 300 ft. - For a range of 3,000 yards the 3-inch gun is elevated at an angle of 5° 5.3' above the line joining gun and target. When fired the projectile actually leaves the gun at an angle of 92.3 mils about this line. How much is the former angle increased by the "jump" of the gun as it is fired?

Ans. 1.8 mils. - A scout measures the angle between a line to an object C and the straight road along which he is passing. He proceeds along the road until, 1,300 yards farther, a line to C makes with the road an angle twice as great. How far is he then from C?

Ans. 1,300 yds.Note. This is the method used by sailors in "doubling the angle on the bow."

- A reconnaissance officer has an accurate map of the surrounding region but does not know his exact position on the map. He sees two objects A and B, which are shown on the map. He measures the angle between the two points and draws lines through A' and B', the map positions of the two points, to meet at the angle found at some point C' on the map. Show that the true position is on the circle drawn through A', B' and C'. If he recognizes a third point whose map position is known he can get a second circle on which his position lies. His true position is at the intersection of these circles.
Note. This is the method of " Italian Resection " used to locate a position with a plane table.

- In maps based on English units 1 in. to the mile, 3 in. to the mile, 6 in. to the mile, and 12 in. to the mile, are commonly used scales. Find the representative fractions of these maps?

Ans. (a) 1/63360, (b) 1121120, (c) 1/10560, (d) 115280. - Find the area of a shrapnel pattern 100 yds. long and 35 yds wide.

Ans. 2,749 sq. yds. - Find the danger space for a horse 15 hands high when the trajectory makes an angle of 252 mils with the ground.

Ans. 20 ft. - An observer 2,000 yds. from the battery and 3,540 yards from the target finds the angle between the battery and the target to be 114 degrees. Find the range from the battery to the target.

Ans. 4,721 yds. - Find the range and deflection components of a wind of ten miles per hour blowing from the southwest if the target is due east of the battery.

Ans. range component = deflection component = 7.1 mi. per hr. - How far is the B. C. station from the right gun if the wheel of the gun carriage, which is 56 in. in diameter, subtends an angle of 4 mil?

Ans. 390 yds. - Of ten shots fired at a range of 2,500 yds eight are to the right of the target and two to the left of it. What change should be made in the deflection?

Ans. Increase 7 mils.

Albert A Bennett.

University of Texas,

May, 1919.